The mathematics of shock reflection-diffraction and von Neumann's conjectures
This book offers a survey of recent developments in the analysis of shock reflection-diffraction, a detailed presentation of original mathematical proofs of von Neumann's conjectures for potential flow, and a collection of related results and new techniques in the analysis of partial differenti...
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| Format: | eBook |
| Language: | English |
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Princeton, NJ
Princeton University Press
2018
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| Series: | Annals of mathematics studies
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| Collection: | DeGruyter MPG Collection - Collection details see MPG.ReNa |
Table of Contents:
- I. Shock reflection-diffraction, nonlinear conservation laws of mixed type, and von Neumann's conjectures
- Shock reflection-diffraction, nonlinear partial differential equations of mixed type, and free boundary problems
- Mathematical formulations and main theorems
- Main steps and related analysis in the proofs of the main theorems
- II. Elliptic theory and related analysis for shock reflection-diffraction
- Relevant results for nonlinear elliptic equations of second order
- Basic properties of the self-similar potential flow equation
- III. Proofs of the main theorems for the sonic conjecture and related analysis
- Uniform states and normal reflection
- Local theory and von Neumann's conjectures
- Admissible solutions and features of problem 2.6.1
- Uniform estimates for admissible solutions
- Regularity of admissible solutions away from the sonic arc
- Regularity of admissible solutions near the sonic arc
- Iteration set and solvability of the iteration problem
- Iteration map, fixed points, and existence of admissible solutions up to the sonic angle
- Optimal regularity of solutions near the sonic circle
- IV. Subsonic regular reflection-diffraction and global existence of solutions up to the detachment angle
- Regularity of admissible solutions near the sonic arc and the reflection point
- Existence of global regular reflection-diffraction solutions up to the detachment angle
- V. Connections and open problems
- The full Euler equation and the potential flow equation
- Shock reflection-diffraction and new mathematical challenges.